One of the oil industry's basic tools for in-situ determination of hydrocarbon saturation is electrical resistivity logging. Since 1942, Archie's empirical relations have been used to calculate oil (and water) saturations in clean sands. In shaly sands the exchange counterions associated with the clay minerals increase the rock conductivity compared to that of a clean or clay-free sand and the simple Archie relations are no longer valid. In the case of shaly sands, the Waxman-Smits equation has been successful in accounting for the additional clay conductance and thereby permitting the quantitative evaluation of oil (and water) saturations in these formations.
The Waxman-Smits equation for 100 percent water-saturated sands refers to the following equation: EQU C.sub.I =1/F*(C.sub.w +BQ.sub.v) (1)
where
C.sub.I =in-phase conductivity (mho-cm.sup.-1) of the completely water-saturated formation PA1 F*=formation resistivity factor as defined by Waxman-Smits PA1 C.sub.w =conductivity of saline solution (mho-cm.sup.-1) contained in the formation rock PA1 Q.sub.v =Waxman-Smits shaliness factor, defined as the cation exchange capacity of the shaly sand per unit pore volume of the sand (meq-ml.sup.-1 or equivalent-liter.sup.-1). PA1 B=equivalent conductance of the exchange cations associated with the clay minerals in the sand formation (mho-cm.sup.2 -meq.sup.-1). B is expressed by Waxman-Smits as a function of C.sub.w. PA1 n*=saturation exponent defined by Waxman-Smits PA1 S.sub.w =fraction of sand pore volume filled with water or the water saturation. Note that S.sub.w =(1-S.sub.o) where PA1 S.sub.o is the fraction of sand pore volume filled with oil or the oil saturation.
F* according to Waxman-Smits is described by the relation: EQU F*=.phi..sup.-m* ( 2)
where .phi. is the porosity of the rock and m* is the cementation factor, usually varying from about 1.5 to 2.2. This equation and a description of its use was published by the authors Waxman-Smits in the 1968 Society of Petroleum Engineering Journal, pages 107-122, June issue, in an article entitled "Electrical Conductivity in Oil-Bearing Shaly Sands".
The Waxman-Smits equation for the in-phase conductivity of a partially brine-saturated shaly sand is: ##EQU1## where C.sub.I '=in-phase conductivity (mho-cm.sup.-1) of the partially oil-saturated shaly sand
From the equations describing C.sub.I and C.sub.I ', the expression for the Resistivity Index I as given by Waxman-Smits is: ##EQU2## Laboratory measurements by Waxman-Thomas as well as current industry usage have confirmed the Waxman-Smits equations for C.sub.I, C.sub.I ' and I as given above. Waxman and Thomas also give the temperature dependence of B. The Waxman-Thomas work was published in the 1974 Journal of Petroleum Technology, pages 213-225, and entitled "Electrical Conductivities in Shaly Sands. I. Relationship between Hydrocarbon Saturation and Resistivity Index. II. The Temperature Coefficient of Electrical Conductivity".
As currently used by the industry, the Waxman-Smits equation requires independent measurement of petrophysical parameters including the cation exchange capacity of the rock per unit pore volume (Q.sub.v). With known techniques it has not been possible to measure this quantity in situ.
Determination of Q.sub.v values generally requires the use of expensive rock samples from the earth formations of interest, either obtained from cores or side-wall samples. Such rock samples are not usually available. Another disadvantage obtaining Q.sub.v from core samples is that the sample may not be representative of the formation as a whole. Furthermore, even if Q.sub.v values are known at the specific depths where samples were taken, calculation of oil saturations are subject to large errors if the in-situ waters are fresh, i.e., contain only low concentrations of soluble electrolytes.
The present invention provides an apparatus and a method using electrical resistivity logging and particularly induced polarization logging to determine the value of Q.sub.v in-situ and the oil/water saturations, S.sub.o and S.sub.w, in clay-bearing sands. The term "induced polarization logging" is used to describe a logging method wherein an electrical current is induced in the formation and then the resulting out-of-phase voltage is measured. In particular, induced polarization logging relates to measurements of the quadrature or reactive component of the electrical impedance of the formation. The history of induced polarization logging is summarized in papers entitled "Complex Formation Resistivity-The Forgotten Half of the Resistivity Log" by Snyder et al. presented at the 18th Annual Logging Symposium, June 5-8, 1977 of the SPWLA and "Investigation of Wells by the Induced Polarization Method (Electrolytic Well Logging)", by Dakhnov et al. in Sb. Promislovaya Geofizika, Vnetoneft; translated by G. V. Keller: The Log Analyst, Nov.-Dec. 1967, pp. 46-82.
In addition to the above papers, Pat. No. 3,902,113 describes an apparatus for obtaining an induced electrical polarization log of an earth formation. In particular, the patent describes an apparatus which includes means for inducing electrical polarization of an earth formation in a manner that each succeeding polarization is in an opposite direction to the preceding one. A measuring circuit controlled by timing logic measures the induced electrical potential difference between two locations during two other time intervals in each cycle of operation. The measured signals are applied to a differential amplifier which produces a difference signal and during one of the two measuring time intervals of each cycle the difference signal is inverted. The signal which is not inverted and the inverted signal are integrated to provide an output which is a measure of the decay of the potential difference. From the above brief description it is seen that the patent applies a DC pulse to the formation, then measures the decay signal to determine the induced polarization of the formation. The decay signal is, of course, the result of the reactive component of the induced polarization and is related thereto.
U.S. Pat. No. 3,895,289 describes another method for determining the electrical resistivity of a shaly formation wherein one measures the dielectric constant of the formation from the voltage decay. Previously determined correlations between dielectric constants and conductivity parameters from earth samples are used to determine the effect of shaliness on resistivity.
As seen from the above brief description of the prior art, none of the prior art measures the quadrature conductivity of the formation at discrete frequencies nor do they provide means for obtaining the shaliness factor Q.sub.v downhole without earth samples. The advantages of measuring the quadrature conductivity at discrete frequencies, rather than from the voltage decay following a pulse, will appear hereafter in the detailed description which follows. In addition, none of the references attempt to define the oil saturation of the formation which is, of course, the most important information that is obtained from logging measurements.